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## Publications

Publications (492)

Goodman proved that the sum of the number of triangles in a graph on n n nodes and its complement is at least n3∕24 n 3 \unicode{x02215} 24 ; in other words, this sum is minimized, asymptotically, by a random graph with edge density 1/2. Erdős conjectured that a similar inequality will hold for K4 K 4 in place of K3 K 3 , but this was disproved by...

We generalize subgraph densities, arising in dense graph limit theory, to Markov spaces (symmetric measures on the square of a standard Borel space). More generally, we define an analogue of the set of homomorphisms in the form of a measure on maps of a finite graph into a Markov space. The existence of such homomorphism measures is not always guar...

In this paper we touch upon three phenomena observed in real life as well as in simulations; in one case, we state mathematical results about the appearance of the phenomenon on arbitrary graphs (networks) under rather general conditions. We discuss a phenomenon of critical fluctuations, demonstrating that an epidemic can behave very differently ev...

Significance
Initial seeding of an epidemic from the best-connected nodes of a network would intuitively lead to the largest outbreak. We challenge this picture and explore a switchover phenomenon: Epidemics started from the central part of a geometric metapopulation network can reach more individuals only if the basic reproduction number is small,...

It is a fundamental question in disease modelling how the initial seeding of an epidemic, spreading over a network, determines its final outcome. Research in this topic has primarily concentrated on finding the seed configuration which infects the most individuals. Although these optimal configurations give insight into how the initial state affect...

Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the "middle ranges", the notion of subgraph densities in these limit objects remains elusive. We define subgraph densiti...

The theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable space...

The theory of graph limits is only understood to any nontrivial degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a station...

Graphings are special bounded-degree graphs on probability
spaces, representing limits of graph sequences that are convergent in a local or
local-global sense. A graphing is compact, if the underlying space is a compact
metric space, the edge set is closed and nearby points have nearby graph neighborhoods.
We prove some simple properties of compact...

Goodman proved that the sum of the number of triangles in a graph on $n$ nodes and its complement is at least $n^3/24$; in other words, this sum is minimized, asymptotically, by a random graph with edge density $1/2$. Erd\H{o}s conjectured that a similar inequality will hold for $K_4$ in place of $K_3$, but this was disproved by Thomason. But an an...

Graphings are special bounded-degree graphs on probability spaces, representing limits of graph sequences that are convergent in a local or local-global sense. We describe a procedure for turning the underlying space into a compact metric space, where the edge set is closed and nearby points have nearby graph neighborhoods.

We exhibit an analogy between the problem of pushing forward
measurable sets under measure preserving maps and linear relaxations in combinatorial
optimization. We show how invariance of hyperfiniteness of graphings
under local isomorphism can be reformulated as an infinite version of a natural
combinatorial optimization problem, and how one can pr...

We study relations between geometric embeddings of graphs and the spectrum of associated matrices, focusing on outerplanar embeddings of graphs. For a simple connected graph G = (V, E), we define a “good” G-matrix as a V × V matrix with negative entries corresponding to adjacent nodes, zero entries corresponding to distinct nonadjacent nodes, and e...

We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on $[0,1]^2$. This leads to a generalization of dense graph limit theory to sparse graph se...

We call a graph positive if it has a nonnegative homomorphism number into any target graph with real edge weights. The Positive Graphs Conjecture offers a structural characterization: these are exactly the graphs that can be obtained by gluing together two copies of the same graph along an independent set of vertices. In this talk I will discuss ou...

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. A special case of this result will assert that if the second and third eigenvalues of the Laplacian are at least $\varepsilon$ apart, then the subgraphs induced by the positive and negative suppo...

We study relations between geometric embeddings of graphs and the spectrum of
associated matrices, focusing on outerplanar embeddings of graphs. For a simple
connected graph $G=(V,E)$, we define a "good" $G$-matrix as a $V\times V$
matrix with negative entries corresponding to adjacent nodes, zero entries
corresponding to distinct nonadjacent nodes...

We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions. In particular it makes it possible to assign limit objects to multigraph sequences...

We study the automorphism group of graphons (graph limits). We prove that
after an appropriate "standardization" of the graphon, the automorphism group
is compact. Furthermore, we characterize the orbits of the automorphism group
on $k$-tuples of points. Among applications we study the graph algebras defined
by finite rank graphons and the space of...

The colored neighborhood metric for sparse graphs was introduced by B. Bollobás and O. Riordan [Random Struct. Algorithms 39, No. 1, 1–38 (2011; Zbl 1223.05271)]. The corresponding convergence notion refines a convergence notion introduced by I. Benjamini and O. Schramm [Electron. J. Probab. 6, Paper No.23, 13 p. (2001; Zbl 1010.82021)]. We prove t...

Mathematical activity has changed a lot in the last 50 years. Some of these changes, like the use of computers, are very visible and are being implemented in mathematical education quite extensively. There are other, more subtle trends that may not be so obvious. We discuss some of these trends and how they could, or should, influence the future of...

Als ich zwischen 1963 und 1966 an der IMO teilnahm, gab es keine Aufgaben aus der Graphentheorie. Heutzutage tauchen sie hingegen recht häufig auf. Woran liegt das? Welche Rolle spielt Graphentheorie in der heutigen Mathematik? Zur Beantwortung dieser Fragen will ich ein paar der vielen Verbindungen zwischen Graphentheorie und anderen Bereichen der...

Paul Erdös was one of the most influential mathematicians of the twentieth century, whose work in number theory, combinatorics, set theory, analysis, and other branches of mathematics has determined the development of large areas of these fields. In 1999, a conference was organized to survey his work, his contributions to mathematics, and the far-r...

The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence). Under appropriate conditions, the...

We consider sequences of graphs (G n ) and define various notions of convergence related to these sequences including “left-convergence,” defined in terms of the densities of homomorphisms from small graphs into G n , and “right-convergence,” defined in terms of the densities of homomorphisms from G n into small graphs. We show that right-convergen...

In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this paper we show that each...

We study "positive" graphs that have a nonnegative homomorphism number into
every edge-weighted graph (where the edgeweights may be negative). We
conjecture that all positive graphs can be obtained by taking two copies of an
arbitrary simple graph and gluing them together along an independent set of
nodes. We prove the conjecture for various classe...

The colored neighborhood metric for sparse graphs was introduced by
Bollob\'as and Riordan. The corresponding convergence notion refines a
convergence notion introduced by Benjamini and Schramm. We prove that even in
this refined sense, the limit of a convergent graph sequence (with uniformly
bounded degree) can be represented by a graphing. We stu...

A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory...

Recently an O∗(n4) volume algorithm has been presented for convex bodies by Lovász and Vempala, where n is the number of dimensions of the convex body. Essentially the algorithm is a series of Monte Carlo integrations. In this paper we describe a computer implementation of the volume algorithm, where we improved the computational aspects of the ori...

A graph G is a k-dot product graph if there exists a vector labelling u:V(G)→ℝ k such that u(i) T u(j)≥1 if and only if ij∈E(G). C. M. Fiduccia et al. [Discrete Math. 181, No. 1–3, 113–138 (1998; Zbl 0974.05058)] asked whether every planar graph is a 3-dot product graph. We show that the answer is “no”. On the other hand, every planar graph is a 4-...

Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by Lov\'asz and Szegedy. In this paper we use this framework to study one of the motivating class of examples, namely randoml...

We prove inequalities between the densities of various bipartite subgraphs in signed graphs. One of the main inequalities is that the density of any bipartite graph with girth 2r cannot exceed the density of the 2r-cycle. This study is motivated by the Simonovits-Sidorenko conjecture, which states that the density of a bipartite graph F with m edge...

How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our
science are quite different. But on a deeper level, there is a more solid connection than meets the eye.

We characterize which graph parameters are partition functions of a vertex model over an algebraically closed field of characteristic 0 (in the sense of [P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, J. Combin. Theory Set. B 57 (1993) 207-227]). We moreover characterize when the vert...

Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an element of a compact second-countable Hausdorff space K. The "local structure" of these objects can be explored...

We study an analogue of the classical moment problem in the framework where moments are indexed by graphs instead of natural numbers. We study limit objects of graph sequences where edges are labeled by elements of a topological space. Among other things we obtain strengthening and generalizations of the main results of previous papers characterizi...

Consider a finite irreducible Markov chain on state space S with transition matrix M and stationary distribution π. Let R be the diagonal matrix of return times, R ii = 1/π i . Given distributions σ, τ and k ∈ S , the exit frequency x k (σ, τ) denotes the expected number of times a random walk exits state k before an optimal stopping rule from σ to...

We prove inequalities between the densities of various bipartite subgraphs in signed graphs and graphons. One of the main inequalities is that the density of any bipartite graph with girth r cannot exceed the density of the r-cycle. This study is motivated by Sidorenko's conjecture, which states that the density of a bipartite graph F with m edges...

We introduce a new approach to constructing networks with realistic features. Our method, in spite of its conceptual simplicity (it has only two parameters) is capable of generating a wide variety of network types with prescribed statistical properties, e.g., with degree or clustering coefficient distributions of various, very different forms. In t...

We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph...

For any two graphs F and G, let hom(F,G) denote the number of homomorphisms F→G, that is, adjacency preserving maps V(F)→V(G) (graphs may have loops but no multiple edges). We characterize graph parameters f for which there exists a graph F such that f(G)=hom(F,G) for each graph G.The result may be considered as a certain dual of a characterization...

The 25th birthday of the LLL-algorithm was celebrated in Caen from 29th June to 1st July 2007. The three day conference kicked off with a historical session of four talks about the origins of the algorithm. The speakers were the three L’s and close bystander Peter van Emde Boas. These were the titles of their talks.
A tale of two papers – Peter van...

Motivated by the result that an evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We sho...

Pivoting, i.e. exchanging exactly one element in a basis, is a fundamental step in the simplex algorithm for linear programming.
This operation has a combinatorial analogue in matroids and greedoids. In this paper we study pivoting for bases of greedoids.
We show that for 2-connected greedoids any basis can be obtained from any other by a (finite)...

In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality...

Freedman, Lovász and Schrijver characterized graph parameters that can be represented as the (weighted) number of homomorphisms into a fixed graph. Several extensions of this result have been proved. We use the framework of categories to prove a general theorem of this kind. Similarly as previous results, the characterization uses certain infinite...

L.G. Khachiyan’s algorithm to check the solvability of a system of linear inequalities with integral coefficients is described.
The running time of the algorithm is polynomial in the number of digits of the coefficients. It can be applied to solve linear
programs in polynomial time.

In this note we show that, for any fixed number r, there exists a polynomial-time algorithm to test whether a given system of linear inequalities Ax≤b is totally dual integral, where A is an integer matrix of rank r.
Key wordsPolynomial-Time–Algorithm–Total Dual Integrality–Hilbert Basis–Integer Linear Programming

In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks: separable elements, with connections (or interactions) between certain pairs of them. These huge networks pose exciting challenges for the mathematician. Graph Theory (the mathematical theory of netwo...

We investigate families of graphs and graphons (graph limits) that are
defined by a finite number of prescribed subgraph densities. Our main focus is
the case when the family contains only one element, i.e., a unique structure is
forced by finitely many subgraph densities. Generalizing results of Turan,
Erdos-Simonovits and Chung-Graham-Wilson, we...

We study generalizations of the "contraction-deletion" relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman, Lovasz and Schrijver, and it relates to their behavior under basic graph operations like contraction and subdivisi...

This book, written by leading experts in combinatorial optimization, features in-depth surveys of current research areas in combinatorial optimization in the broad sense, ranging from applied graph theory to mathematical programming. It also contains numerous new results and shows many interesting current research directions. This book will be indi...

The Distinct Element Method (DEM) is a popular tool to perform granular media simulations. The two key elements this requires are an adequate model for inter-particulate contact forces and an efficient contact detection method. Originally, this method was designed to handle spherical-shaped grains that allow for efficient contact detection and simp...

Graph property testing is the third reincarnation of the same general question, after statistics and learning theory. In its simplest form, we have a huge graph (we don’t even know its size), and we draw a sample of the node set of bounded size. What properties of the graph can be deduced from this sample?
The graph property testing model was first...

To define an exact pressure-flow correlation in the upper urinary tract using an improved measurement method, to quantitatively characterize the degree of postrenal obstruction and to find a simple way of calculating it in everyday urological practice.
The data of 112 cases were included in the analysis. The dynamic method of a multistep, constant...

We prove a general theorem on semigroup functions that implies characterizations of graph partition functions in terms of the positive semidefiniteness (‘reflection positivity’) and rank of certain derived matrices. The theorem can be applied to undirected and directed graphs as well as hypergraphs.

For a symmetric bounded measurable function W on [0, 1]2 and a simple graph F, the homomorphism density
$$t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$$ can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables.
The main moti...

We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the random graph obtained when using the function values as edge probabilities. We give a characterization of properties testable this way, and extend a numbe...

We prove that if a sequence of graphs has (asymptotically) the same distribution of small subgraphs as a generalized random graph modeled on a fixed weighted graph H, then these graphs have a structure that is asymptotically the same as the structure of H. Furthermore, it suffices to require this for a finite number of subgraphs, whose number and s...

Hungarian mathematics has always been known for discrete mathematics, including combinatorial number theory, set theory and recently random structures, combinatorial geometry as well.
The recent volume contains high level surveys on these topics with authors mostly being invited speakers for the conference "Horizons of Combinatorics" held in Balato...

In many areas of science huge networks (graphs) are central objects of study: the internet, the brain, various social networks,
VLSI, statistical physics. To study these graphs, new paradigms are needed: What are meaningful questions to ask? When are
two huge graphs “similar”? How to “scale down” these graphs without changing their fundamental stru...

A flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge. In this article, we study single-commodity confluent flow problems, where we need to route given node demands to a single destination using a confluent flow. Single- and multi-commodity confluent flows arise in a variety of applica...

The class of logconcave functions in ℝn is a common generalization of Gaussians and of indicator functions of convex sets. Motivated by the problem of sampling from logconcave density functions, we study their geometry and introduce a technique for “smoothing” them out. These results are applied to analyze two efficient algorithms for sampling from...

Szemerédi’s regularity lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph
property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi’s lemma can be thought
of as a result in analysis. We show three different analytic interpretations.

We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences: “left convergence” defined in terms of the densities of homomorphisms from small graphs into Gn; “right convergence” defined in terms of the densities of homomorphisms from Gn into small graphs; and convergence in a suitably defined metric.In...

We propose a scheme for building peer-to-peer overlay networks for broadcasting using network coding. The scheme addresses
many practical issues such as scalability, robustness, constraints on bandwidth, and locality of decisions. We analyze the
system theoretically and prove near optimal bounds on the parameters defining robustness and scalability...

This article continues the study of multiple blocking sets in PG(2, q). In [3], using lacunary polynomials, it was proven that t-fold blocking sets of PG(2, q), q square, t < q(1/4)/2, of size smaller than t(q + 1) + c(q)q(2/3), with c(q) = 2(-1/3) when q is a power of 2 or 3 and c(q) = 1 otherwise, contain the union of t pairwise disjoint Baer sub...

This chapter introduces a matrix, the kth connection matrix, for a given graph parameter and integer k ≥ 0. The properties of these matrices are closely connected with properties of the parameter. For example, the rank of this matrix is considered the minimum number of real numbers that has to be communicated across a node-cut of size k in order to...

: A matching in a graph G is a set of lines, no two of which share a common point. A matching is perfect if it spans V(G). The problem of finding a matching of maximum cardinality in a graph models a number of significant real-world problems and, in addition, is of considerable mathematical interest in its own right. Matchings are in a sense among...

We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object,” namely a symmetric measurable function . This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object....

A Halin graph is a plane graph H = T U C, where T is a plane tree with no vertex of degree two and at least one vertex of degree three or more, and C is a cycle connecting the endvertices of T in the cyclic order determined by the embedding of T. We prove that such a graph on n vertices contains cycles of all lengths l, 3 ≤ ln, except, possibly, fo...

We prove that the hit-and-run random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of Lovasz and Vempala (2004), where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we derive asymptotically faster algor...

Connection matrices were introduced in [M. Freedman, L. Lovász, A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs (MSR Tech Report # MSR-TR-2004-41) ftp://ftp.research.microsoft.com/pub/tr/TR-2004-41.pdf], where they were used to characterize graph homomorphism functions. The goal of this note is to determine the ex...

The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of t...

A hypergraph is called normal if the chromatic index of any partial hypergraph H' of it coincides with the maximum valency in H'. It is proved that a hypergraph is normal iff the maximum number of disjoint hyperedges coincides with the minimum number of vertices representing the hyperedges in each partial hypergraph of it. This theorem implies the...

Linear programming has been one of the most fundamental and successful tools in optimization and discrete mathematics. Its applications include exact and approximation algorithms, as well as structural results and estimates. The key point is that linear programs are very efficiently solvable, and have a powerful duality theory.

We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f1, f2 defined on the subsets of a finite set S, satisfying ∑X⊆S fi(X) ≥ 0 for i ∈ {1, 2}, there exists a positive multiplicative set function μ over S and two subsets A, B ⊆ S such that f...

A hypergraph is called normal if the chromatic index of any partial hypergraph H′ of it coincides with the maximum valency in H′. It is proved that a hypergraph is normal iff the maximum number of disjoint hyperedges coincides with the minimum number of vertices representing the hyperedges in each partial hypergraph of it. This theorem implies the...

We consider the problem of fairly matching the left-hand vertices of a bipartite graph to the right-hand vertices. We refer to this problem as the optimal semi-matching problem; it is a relaxation of the known bipartite matching problem. We present a way to evaluate the quality of a given semi-matching and show that, under this measure, an optimal...

We present a new algorithm for computing the volume of a convex body in Rn. The main ingredients of the algorithm are (i) a “morphing” technique that can be viewed as a variant of simulated annealing and (ii) a new rounding algorithm to put a convex body in near-isotropic position. The complexity is O*(n4), improving on the previous best algorithm...

Motivated by applications in combinatorial optimization, we initiate a study of the extent to which the global properties of a metric space (especially, embeddability in l(1) with low distortion) are determined by the properties of small subspaces. We note connections to similar issues studied already in Ramsey theory, complexity theory (especially...

Proving integrality gaps for linear relaxations of NP optimization
problems is a difficult task and usually undertaken on a
case-by-case basis. We initiate a more systematic approach. We
prove an integrality gap of $2 -o(1)$ for three families of linear
relaxations for VERTEX COVER, and our methods seem
relevant to other problems as well.